Gerunds (or verbal nouns) look like present participles, but they function as nouns. A gerund can be the subject of a sentence, an object, the object of a preposition, a subject complement, or the complement of a possessive adjective. Gerunds can answer the question what.
Now, let's start with your second question:
Can you say, what is the difference between "This bad drawing of a dog
is not acceptable for your project" and "This bad draw of a dog is not
acceptable for your project"?
First, drawing is not a gerund in your example sentence. It is a just a regular noun.
This sentence contains a gerund:
Drawing is fun.
To answer your second question: the first sentence is correct, but the second sentence is not. A "draw" is not a picture.
On to your first question, which is more complex:
You have asked when we should use a gerund inside a compound noun vs a regular noun. Right. Let's look at examples and compare them:
battery charging efficiency
battery charge efficiency
Regrettably, your uncle knows nothing about battery charging
(how to charge batteries efficiently--the process)
However, your aunt knows a lot about battery charge efficiency.
(facts about the efficiency of battery charges)
Notice that these are two very different things: battery charging efficiency and battery charge efficiency. Tᴚoɯɐuo rightly says that the gerund describes a process and "contains a tad more information".